Up: Pole coordinates and shape
In order to determine the orientation of the rotational axis and the shape
of the asteroid, we used the amplitude-magnitude (AM) method suggested
by Zappalà (1981) and
refined by Zappalà et al. (1983a), which is based
on the assumed ellipsoidal shape of the asteroid (with semi-axes
a > b > c) and on the relationships between the aspect angle, the lightcurve
amplitude and the asteroid magnitude at the lightcurve maximum, all obtained in
several oppositions (at least three). It is important to note that usually
the real shape of asteroids is different from the ellipsoidal
one and moreover the albedo is often not homogeneous over the entire
surface. These discrepancies often lead to conflicting results especially
when the data are few.
From the lightcurves, we obtain the magnitude V at the maximum of lightcurve
and the amplitude A, depending on the rotation axis orientation
and on the ratio of the maximum to minimum cross-sections of the
asteroid, respectively.
If we assume the smaller axis c to be the asteroid rotation axis, the ratio
between the two other axes, and subsequently their single values, can
be obtained from the plot,
if we have a continuous and good distribution in longitude of the observed
amplitudes. The modelling curves were obtained using the least square
method.
In some cases the extrema of the theoretical curves seem to be
overestimated with respect to the observed values. This fact depends on
the computing program that, in the absence of observed values at the
longitudes of the maximum or the minimum, takes into account the slope
of the ascending or descending branches.
From the axes ratios it is possible to obtain the value of the aspect angle
(with an uncertain definition of the north or south pole) and
hence the pole longitude.
Table 1:
References of the lightcurves used for the estimation of the V
magnitudes and for the construction of the plots. The symbol
before the author's name is the same used in the corresponding
plots
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Table 1:
continued
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Following Zappalà et al. (1990) suggestions, we have corrected the
lightcurve amplitude for its dependence on the phase angle, by means of the
relationship where is the observed lightcurve amplitude, is
the solar phase angle and m is a coefficient depending on the asteroid
taxonomic class. The of each asteroid was computed
adopting the value
, the arithmetic average of all phase angles.
In Table 1, for each asteroid, the references of the lightcurves used for the
estimation of the V magnitude and for the construction of the
plots are reported. The symbol before the author's name is that used in
the corresponding plots. Only lightcurves at least 90%
covered were utilized. Due to the available lightcurves, their minimum number
(at least three) necessary for applying the (AM) method and to their
distribution in longitude, it was possible to compute the pole coordinates
and the axes ratios only for 30 asteroids. In
Fig. 1, using different
symbols for different authors as indicated in Table 1, the
plots of these asteroids are reported. The adopted values are the
mean values computed over the duration of each lightcurve.
The filled symbols indicate the observed values of the amplitude A, the
empty ones the corresponding values at longitudes , the
continuous and dashed
(in the case of two solutions) lines the theoretical curves.
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Figure 1:
Amplitude-longitude plots of the asteroid to
which it was possible to apply the (AM) method. The filled symbols, as
reported in Table 1, indicate the observed values of the amplitude A,
the empty ones the corresponding ones at longitudes , the continuous and (in the case of two solutions) dashed lines the
theoretical curves |
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Figure 1:
continued |
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Figure 1:
continued |
Up: Pole coordinates and shape
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